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Something to Think About. What are good activities for helping students to learn their basic facts?. Secondary Numeracy Project. Fraction, Decimal, Percents. Fraction Constructs. Fractions are needed when whole units are inadequate to get a job done. Part-Whole: continuous & discrete.

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Something to Think About

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Something to Think About

What are good activities for helping students to learn their basic facts?

Secondary Numeracy Project

Fraction,

Decimal,

Percents

Fraction Constructs

Fractions are needed when whole units are inadequate to get a job done.

Part-Whole: continuous & discrete

Continuous fraction model

Fractions are met as cutting up an entire object

e.g. one quarter of an apple

This single object soon extends to multiple objects to be shared

e.g. sharing two apples between 5 people

Leads to the concept that fractions are divisions

e.g.

Discrete fraction model

Fractions are met through sharing a collection of objects

e.g. 4 kids share 12 cookies

Problems initially lead to discrete, whole number answers

Later problems merge with the continuous as “remainder” objects are cut

e.g. 4 kids share 10 cookies

Handout: A Teaching Progression for Fractions

Stages – Continous & Discrete

Any surprises?

Look at Knowledge & Strategy Tests

Knowledge Test: Fractions

Stage 4 (3 sec)

9: Write one half as a fraction.

10: Write one sixth as a fraction

11: Write one third as a fraction

12: Write one quarter as a fraction

Stage 5 (5 sec)

13: Which of these fractions is the smallest?

14: Which of these fractions is the largest?

Stage 6 (5 sec)

15: Which of these numbers is the same as eight-sixths?

Stage 7 (5 sec)

16: Which of these fractions is the same as two-thirds?

17: Which of these fractions is the same as three-quarters?

Stage 8

18: Which of these fractions is the smallest? (10 sec)

19: Which of these fractions is the largest? (10 sec)

Strategy Test: Fractions

Stage 2-4 (materials)/Stage 5 (facts):

This cake has been cut into thirds. Here are twelve jellybeans to spread out evenly on top of the cake. You eat one third of the cake. How many jellybeans do you eat?

Stage 6:

What is of 28?

Stage 7:

12 is of a number. What is the number?

Stage 8:

There are 21 boys and 14 girls in Ana’s class. What percentage of Ana’s class are boys?

It takes 10 balls of wool to make 15 beanies. How many balls of wool does it take to make 6 beanies?

SNP Teaching Model

Use a concrete representation (Materials/Diagrams)

Encourage imaging and visualisation (Imaging)

Push to the inherent property and generalisation (Abstraction)

How can we use this with fractions?

Fraction Strips

Work in groups of 8

To start each person has a different colour paper

Fold paper in eighths – make sure everyone folds the same direction so that all eighths are the same size

Cut 1/8 strips (along fold lines)

Share out your strips so that everyone ends up with 8 strips of eight different colours

Cut strips as directed to match magnets

Uses of fraction strips

Order unit fractions and fractions with the same denominator and explain why they are larger or smaller

Which is bigger? Why?

Order fractions visually using materials, including improper fractions and explain what the numerator and denominator mean.

Make pairs of fractions. Which is bigger?

Which is the bigger fraction?

Many students miss the comparative nature of fractions – the relationship between the numerator and denominator. (Some research suggests that the failure to understand this is a reason why students have difficulty with fractions)

Complexifying the constructs

What fraction is shaded pink?

What fraction is shaded pink?

3

4

48

64

48 = 3 x 16

64 = 4 x 16

Operational Construct

¼ of 36

halve and halve again

relationship between times/division and fractional strategies

¾ of 36

0.25 of 36

25% of 36

Book 7

Changing up the way the question is asked.

A Teaching Progression for Percentages

Requires high levels of thinking

Decimals (decimal fractions)

Other handouts

Maths is......explaining...

Remember the long term goal – improving their algebraic thinking:

One thing you can do to help this every day is to ask questions in class: